Regional Business Review, Vol. 24, May 2005, pp.50-75

Firm Value And The Debt-Equity Choice Professor Rob Hull, School of Business, Washburn University

Abstract Building on the no growth perpetuity framework first developed by Modigliani and Miller (1963), this paper attempts to offer gain to leverage (GL) formulations useable by managers in making debt-equity choices. These formulations focus on how changes in equity and debt discount rates influence firm value. A real world application (using data suggested by independent analysts) seeks to determine the gain to leverage for different debt-equity choices. Using our formulation with constant growth, we offer results that can support the suggested target debt-equity choice as the choice that maximizes firm value.

I. Introduction and Background According to Compustat, since the beginning of the century there have been about 1,650 firms per year that on average have reported no long-term debt (which includes capitalized lease obligations). Gopalakrishnan (1994) indicates about 30 percent of such unlevered firms will issue debt within a year and maintain it for a prolonged (if not permanent) period of time. However, larger firms without long-term debt are a rarity as shown by Agrawal and Nagarajan (1990) who find only 104 such firms listed on major U.S. stock exchanges. This suggests that most managers, at least for larger firms, behave as if value can be added by choosing some positive debt level when financing their operating assets. Theoreticians offer various formulas to support the managerial decision to issue debt. The forerunner of this line of research is Modigliani and Miller (1963), referred to as MM. They derive a gain to leverage (GL) formulation in the context of an unlevered firm issuing risk-free debt to replace risky equity. For MM, GL is the corporate tax rate multiplied by debt value. The applicability of MM’s GL formulation is limited as it implies that financial executives issue unrestricted amounts of debt. Extensions of MM consider a variety of leverage-related wealth effects (most noteworthy, the 50 Northwest Missouri State University

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effects stemming from personal tax, flotation costs, bankruptcy, agency, and asymmetric information considerations). Empirical researchers offer differing opinions concerning the strength of leverage-related effects. While early researchers (Warner, 1977; Miller, 1977) suggest such effects may be unimportant (at least for larger firms), later investigators (Altman, 1984; Cutler and Summers 1988) contend otherwise indicating such effects would be significant if quantified. Graham and Harvey (2001) offer support for leverage-related effects but restrict this support by noting there is little evidence that executives are concerned about some effects (namely, personal taxes, transactions costs, asset substitution, free cash flows, and asymmetric information). Regardless of the significance of leverage-related effects, some researchers (Graham and Harvey, 2001; Pinegar and Wilbrecht, 1989) indicate that firms may be more concerned with an amount of debt that gives flexibility for future opportunities. Other researchers (Fischer, Heinkel, and Zechner, 1989; Kayhan and Titman, 2004) downplay the need for debt flexibility by offering evidence for the role performed by tax and bankruptcy cost effects. Hull (1999) presents event study evidence consistent with leverage-related effects determining an optimal debt level. Given the presence of debt in the capital structure of most firms as well as the empirical evidence concerning leverage-related effects, there is a need to offer usable equations that can quantify these effects. This paper aims to fill this void by offering G L formulations that quantify leverage-related wealth effects. This is done through perpetuity GL formulations that make explicit how changes in equity and debt discount rates impact firm value. To the extent changes in such discount rates can be accurately estimated along with values for other relevant variables (such as growth and tax rates), the GL formulations given in this paper can be used to measure the dollar impact of a proposed capital structure change. Consequently, it is possible for financial executives to make a debt-equity choice that maximizes firm value. The remainder of the paper is organized as follows. Section II reviews the traditional G L perpetuity formulations. Section III derives GL formulations for an unlevered firm situation (although not shown in this paper, similar but lengthier formulations could be offered for firms that are already levered). Section IV gives computations for an application using real data. Section V reports the application’s results for ten key variables for nine debt-equity choices. Section VI presents limitations of the application and Section VII gives summary statements. 51 Northwest Missouri State University

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II. Traditional Perpetuity Gain to Leverage Formulations This paper’s GL formulations are rooted in and developed within the no growth perpetuity framework of MM (1963) and Miller (1977). This section reviews these GL formulations and their extensions. It ends by indicating the need to incorporate discount rates in GL formulations. A. The MM Gain to Leverage Formulation MM analyze the valuation impact of a debt-for-equity transaction. The simplifying conditions explicitly or implicitly used in their analysis include: (i) two security types (an unlevered firm with risky equity that issues risk-free debt); (ii) only corporate taxes (no personal taxes on income from either equity or debt); (iii) level perpetuities (which can approximate any series of unequal cash flows); (iv) no growth (depreciation each year equals investment to keep the same amount of capital); (v) no imperfections (i.e., no leveraged-related effects such as flotation costs, bankruptcy costs, agency effects, or asymmetric information effects); and, (vi) equivalent return classes (the CAPM had not yet been developed).

Given these conditions, MM argue that GL is the exogenous corporate tax rate (TC) times the value of perpetual risk-free debt (D) such that GL = TCD.

(1)

D is the chosen perpetual interest payment (I) divided by the exogenous cost of capital on risk-free debt (RF). As D increases, MM posit that there is an increase in the rate at which risky equity is discounted. However, no quantitative application is made of any net negative impact on firm value of the increase in equity's discount rate. Similarly, no detailed valuation analysis is made of the GL ramifications if debt is risky. However, if debt is risky, then we have D= I

(2)

RD

where RD > RF with RD now an increasing function of debt. While there are other forms of financing that might affect the debt-equity choice, little attention is given to these forms. For example, one form that might affect the choice is long-term lease financing. However, because any such lease payment acts like debt by lowering the firm’s taxable income and increasing its financial risk, it resembles debt and can be treated as part of D. This is true for any off-balance-sheet items that behave like debt. 52 Northwest Missouri State University

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B. Extensions of the MM Formulation Those extending the GL equation of MM (Baxter, 1967; Kraus and Litzenberger, 1973; Kim, 1978) assume risky debt. They argue that increasing debt levels are associated with increasing bankruptcy costs such that an optimal debt level exists where the negative bankruptcy costs effect offsets the positive tax shield effect. Increasing levels of debt can cause firm value to fall for reasons other than bankruptcy costs. For example, Jensen and Meckling (1976) examine a wider range of leverage-related costs that they call agency costs. Regardless of corporate tax shield and bankruptcy considerations, net agency effects can impact GL. For example, increasing debt can initially cause net gains owing to the reduction in owner-manager monitoring costs, but can eventually lead to net losses due to the escalation in costs caused by restrictive debt covenants. Drawing from the work of Farrar and Selwyn (1967), Miller (1977) assumes personal taxes and extends (1) such that GL = [1 where α =

α]D

(3)

(1 TPE )(1 TC ) with TPE and TPD the personal tax rates applicable to income from equity (1 TPD )

and debt, and D now equals

(1 TPD )I . For Miller, costs related to the increase in debt (in particular, RD

bankruptcy costs) are inconsequential so that the effect of personal taxes alone offset the effect of corporate taxes. For example, Miller argues that α ≈ 1, and thus GL ≈ zero (e.g., GL = [1 1]D ≈ 0). Regardless, as [1 than GL in (1). Even if [1

α]D ≈ [1

α] in (3) takes on values smaller than TC, then GL in (3) becomes less α] = TC , GL in (3) is less than GL in (1) if TPD > 0 since D in (3) is

adjusted for personal taxes and now equals

I (1 TPD )I instead of just . RD RD

Even if Miller is correct, signaling theory (Leland and Pyle, 1977; Ross, 1977; Myers and Majluf, 1984) suggest that an increase in a firm’s debt-to-equity ratio can lead to an increase in firm value. For example, Myers and Majluf (1984) argue that if managers are better informed than outside investors, firms are more likely to retire equity when it is undervalued. Thus, a debt-for-equity transaction would signal positive news in the sense underpriced securities are being retired (in addition to any other positive signal the firm conveys about it future cash flows covering larger 53 Northwest Missouri State University

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interest payments). Empirical research (Copeland and Lee, 1991; Hull and Michelson, 1999) offers evidence consistent with debt-for-equity transactions signaling positive news (including the conveyance of reduced risk as seen in lower betas and thus reduced required rates of return). In conclusion, signaling theory suggests exchanging debt for equity can cause GL > 0 to hold even if there are no other leverage-related effects. Ensuing GL extensions of MM (DeAngelo and Masulis, 1980; Kim, 1982; Modigliani, 1982; Ross, 1985) consider a variety of leverage-related costs and show that an optimal debt level exists even when personal taxes are recognized. Leland and Toft (1996) extend the closed-form results of Leland (1994) to a much richer class of possible debt structures permitting the study of the optimal amount of maturity of debt. Leland (1998) attempts to provide quantitative guidance on the amount and maturity of debt, the financial restructuring, and the optimal risk strategy. For the most part, the GL extensions are characterized by the inability to make explicit how changes in equity and debt discount rates impact firm value within a model that financial managers might find useable.

III. Formulations That Incorporate Discount Rates In this section, practical GL formulations are derived for managers making their debt-equity choices. These equations consider the impact of equity and debt discount rates for an unlevered firm for three situations: (i) no personal taxes and no growth, (ii) personal taxes and no growth, and, (iii) personal taxes and constant growth. A. Gain to Leverage Formulation without Personal Taxes A GL formulation that includes discount rates can be derived from the definition that G L is levered firm value (VL) minus unlevered firm value (VU). We have GL = VL

VU

(4)

where VU and VL are defined below and the general MM conditions described earlier hold. VU is the same as unlevered equity value (EU). EU is the uncertain perpetual after-corporate tax cash flow available to unlevered equity of (1 TC)C divided by the exogenous unlevered equity discount rate (RU). We have VU = EU =

(1 TC )C RU

(5)

where C is the perpetual before-tax cash flow available to unlevered equity owners with RU > RD if 54 Northwest Missouri State University

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the firm should choose to issue debt. Note that C assumes all expenses are cash expenses so that before-tax cash flow equals taxable income. VL is levered equity value (EL) plus debt value (D). EL is the uncertain perpetual after-corporate tax cash flow available to levered equity of (1 TC)(C I) divided by the endogenous levered equity discount rate (RL). We have EL =

(1 TC )( C I) RL

(6)

where RL > RU with RL positively related to debt (e.g., the cash flow to equity owners has more uncertainty as debt increases). Inserting (6) and (2) into the definition VL = EL + D gives VL =

(1 TC )( C I) I + RL RD

(7)

where RD = RF only if debt is risk-free debt (as MM assume or as the CAPM suggests when a debt beta is assumed to be zero, which is often the assumption). Regardless, the derivation of the below GL formulation is unimpeded if RD is endogenously determined by the debt level choice such that RD > RF holds. The GL formulation for an unlevered firm issuing debt can now be derived. After substituting (7) into (4) and noting VU = EU, Appendix A shows GL = 1

RU αR D D+ RL RL

1 EU

(8)

where α = (1 TC). The 1st component, 1

αR D αR D D, is always positive if D > 0 since < 1. If D = 0, then this RL RL

component is zero. The 2nd component,

RU RL

1 EU, is always negative if D > 0 since EU > 0 and

RU < 1. If D = 0, then RU = RL and the 2nd component (like the 1st component) will also be zero RL

when D = 0 holds. Thus, if D = 0 then (8) implies that GL = 0. But if D > 0 then (8) can be either positive or negative depending on which component has the greatest absolute value. One can note that the 1st component is similar to the traditional GL formulations except α is 55 Northwest Missouri State University

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multiplied by a value less than one (e.g.,

RD < 1) causing the component to be more positive than RL

the traditional GL formulations. In looking at the 2nd component, we can see that GL can be viewed as being related to how much the increase in debt negatively affects outstanding equity through the percentage increase in its discount rate. This relationship is consistent with the intuitive notion that as leverage increases risk (and thus the required rate of return) then the value of the firm should fall accordingly.

B. Gain to Leverage Formulation with Personal Taxes and Constant Growth When personal taxes are considered, we can show (in a fashion similar to that found in Appendix A and later in Appendix B) that GL can still be expressed as (8) if definitions for α, VU, EL, and D are modified to incorporate personal tax rates. For example, we still have GL = 1 for (8) only now we have: α =

=

RU αR D D+ RL RL

1 EU

(1 - TPE )(1 TC )C (1 TPD )I (1 TPE )(1 TC ) ; VU = EU = ;D= ; and, EL (1 TPD ) RU RD

(1 TPE )(1 TC )( C I) . For the 1st component to still be positive (when D > 0) is now a bit more RL

complicated. This is because, for

αR D < 1 to now hold, restrictions must be placed on TC, TPE, and RL

TPD (and these restrictions depend on values for RD and RL). Just as the Miller (1977) GL formulation given in (3) reduces to the MM formulation given in (1) if TPE = TPD, so this paper’s GL formulation given in (8) reduces to (1) if RU = RL = RD and TPE = TPD. With definitions for α, VU, EL, and D modified to include personal tax rates, equation (8) reduces to the Miller formulation given by (3) if RU = RL = RD. These reductions reflect the MM derivational procedure that assumes equality of discount rates when denominations (discount rates) are ignored in the factoring process. Appendix B derives a GL equation when both personal taxes and constant growth is considered. Constant growth implies a current dollar change in after-tax cash flows (δg), which we define as δg = (1 TPE)(1 TC)(C I)( γ TL arg et )

(9)

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where γ TL arg et is the growth rate when the firm achieves its targeted (and assumedly desired optimal) amount of interest paid. To derive this GL equation, definitions for VU and EL must be modified as follows: VU = EU =

(1 TPE )(1 TC )C (1 TPE )(1 TC )( C I) and EL = where γU is the growth rate if R U γU R L γL

the firm is unlevered and γL is a growth rate for a given levered situation. For the unlevered growth rate (γU), we have γU =

δg (1 TPE )(1 TC )C

.

(10)

For the levered growth rate (γL), we have γL =

δg

(11)

(1 TPE )(1 TC )( C I)

where γL > γU since C > (C – I). We can note that ceteris paribus γL increases as I increases. Also, γL = γ TL arg et when the target leverage ratio is achieved. With γU as the growth rate for the unlevered situation and γL the growth rate for a given levered situation, Appendix B shows that GL = 1

RU αR D D+ R L γL RL

γU γL

1 EU

(12)

where (12) reduces to (8) if there is no growth such that γL = γU = 0. Note that the 1st component can become negative if αRD > (RL γL) holds, while the 2nd component can become positive if (RU γU) > (RL γL) holds. This can occur for large amounts of debt where γL becomes large causing (RL γL) to become small.

IV. Application Using Company Data This section presents our application, which considers Australian Gas Light Company (AGL Co.), a major retailer of gas and electricity with about three million customers. We attempt to determine GL if the suggested target debt-equity choice is reached and simultaneously try to determine if this is the optimal. To achieve this aim we gather needed market data and company data from independent sources that include a firm offering audit, tax, and advisory services (KPMG International) and one offering brokerage services (State One Stockbroking Ltd.). To compute GL, we will use equation (12) with all monetary values given in Australian dollars (A$). 57 Northwest Missouri State University

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A. Market and Tax Rate Data for Application From http://www.ipart.nsw.gov.au/papers/KPMG_February_04.pdf, we find a 48 page report on AGL Co. where KPMG estimates values for variables that affect AGL Co.’s valuation. In Table 1, we give KPMG’s suggestions (as of February 2004) for market and tax rate data.

Table 1. Market and Tax Rate Data RR = Real Rate = 3.42% RINF = Inflation Rate = 2.17% RF = Risk-Free Rate = RR + RINF + (RR)(RINF) = 3.42% + 2.17% + (3.42%)(2.17%) = 5.6642% MPREM = Market Premium = (RM – RF) = 6.00% TC = Corporate Tax Rate = 30.00% λ = Imputation Tax Credit = 40.00% TE = Effective Tax Rate = TC(1–λ) = 30%(1 – 0.4) = 18.00% TE = Effective Tax Rate =

1 TC 1 [TC (1 λ)]

=

1 0 .3 1 [0.3(1 0.4)]

= 0.14634 or 14.634%

Average TE = (18.00% + 14.634%) = 16.317% ≈ 16.32% Average (1 – TE) = 1 – 0.16317 = 0.83683 ≈ 83.68% TPE = Personal Tax Rate on Equity Income = 4.77% TPD = Personal Tax Rate on Debt Income = 20.34% α=

(1 TPE )(1 TC ) (1 TPD )

=

(1 0.0477)(1 0.3) = 0.83682 (1 0.2034)

≈ 83.68% ≈ Average (1 – TE)

(1 – α) = (1 – 0.83682) = 0.16318 ≈ 16.32% ≈ Average TE KPMG gives no estimates for personal tax rates so we turn to another approach that uses knowledge of imputation credits (λ). Under the Australian imputation tax system, domestic equity investors receive a taxation credit for dividends paid out of after-tax firm returns. In essence, an imputation tax system offsets the corporate tax advantage of debt in a manner analogous to when equity owners have a lower tax rate than debt owners (TPE < TPD). KPMG suggests that λ = 0.4 and that TC (for equity owners) is effectively reduced to a lower rate (TE). As seen in Table 1, using the average of the computations given by KPMG, we get Average TE = 16.317% ≈ 16.32%. Using this 58 Northwest Missouri State University

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value to estimate the personal tax rates, we proceed by noting that α = (1 TE) ≈ 1

0.1632 ≈ 0.8368

or about 83.68%. This value for α can be equated with a number of values for TPE and TPD (when TC = 30%) including the two we use in our application: TPE = 4.77% and TPD = 20.34%. As seen in Table 1, for these two values, α ≈ 83.68%. We should point out that after personal tax values for EU, D, EL and VL when TC = 30%, TPE = 4.77%, and TPD = 20.34% will differ from those when TC (given by TE) = 16.32%, TPE = 0 and TPD = 0. However, if we look at before personal tax values for GL, the reductions in after personal tax values for EU, D, EL and VL caused by using TPE = 4.77% and TPD = 20.34% will be overcome from the standpoint of getting values before lowered by paying personal taxes.

Table 2. Beta and Cost of Capital Data Current Book

D E

=

$3,241,500,000 ≈ 1.0 implies VD = 0.5 and VE = 0.5 $3,153,000,000

DPREM = Debt Premium = βD(RM – RF) = βD(MPREM) = 1.75% βD = Debt Beta = DPREM / MPREM = 1.75% / 6% = 0.291667 RD = Cost of Debt = RF + βDMPREM = 5.66421% + 0.291667(6%) = 7.41421% ≈ 7.41% βL = Levered Equity Beta = 1.05 RL = Cost of Levered Equity = RF + βLMPREM = 5.6642% + 1.05(6%) = 11.9642% ≈ 11.96% βU = Unlevered Equity Beta =

L

[

D

(1 TC )(

1 [(1 TC )(

D E

D E

)]

)]

=

1.05 [0.291667(1 0.3)(1.0)] = 0.737745 1 [(1 0.3)(1.0)]

RU = Cost of Unlevered Equity = RF + βUMPREM = 5.664214% + 0.737745(6%) = 10.0907%

B. Beta and Cost of Capital Data for Application As seen in Table 2, the annual report for AGL Co.’s fiscal year ending June 2004 indicates its current book

D E

ratio ≈ 1.0 where D is total liabilities and E is shareholders’ equity. The values

suggested by KPMG for betas and costs of capital are assumed to correspond with this book debtequity choice of 1.0. To get AGL Co.’s cost of debt, we begin by noting that KPMG estimates a debt margin of 1.75%, which absent other costs suggests a debt premium (DPREM) of 1.75%. As seen in Table 2, this premium implies the debt beta (βD) = 0.2917. Using the CAPM, we get the cost of debt 59 Northwest Missouri State University

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(RD) ≈ 7.41%, which is the midpoint of KPMG’s range estimated at 7.21% to 7.61%. KPMG suggests AGL Co.’s levered beta (βL) = 1.05. Using the CAPM, this value for βL implies its cost of levered equity (RL) ≈ 11.96%. .

KPMG indicates, regardless of any formula chosen, BD should not be assumed zero when D

computing the unlevered beta (βU). Given TC = 30%, the current book leverage ratios ( E = 1.0,

D V

=

0.5, and VE = 0.5), βL = 1.05, and βD = 0.29, Table 2 uses the formula given by Hamada (1972) to get βU = 0.7377. Using the CAPM, this value for βU implies the cost of unlevered equity (RU) = 10.0907%. Given the data in Table 2, we can try to estimate betas (and thus RD’s and RL’s) for various debt level choices. For simplicity, we consider only nine

D V

choices with each choice given in book

values (before market adjustments caused by a positive GL are made). We determine each choice based on the number of shares retired. This is accomplished by noting AGL Co. has outstanding levered shares (NL) of 456,000,000. Given NL and current book

E V

= 0.5, we can estimate the number

of shares if the firm was unlevered (NU). As will be seen in Table 3, we have NU =

NL Current Book

E V

=

456,000,000 = 912,000,000. 0.5

From this NU value, we can get the number of shares retired (SR) for each debt level choice. For example, if

E V

= 0.9, then

D V

D

= 0.1 and AGL Co. would retire SR = ( V )(NU) = 0.1(912,000,000) =

91,200,000 shares. Similarly, we can compute SR for any debt level choice. For each choice faced by an unlevered firm, the value of the debt issued (D) should ceteris paribus equal the dollar amount of the retired shares where D = PU(SR) with PU the unlevered share price (which can be viewed as the current market price minus any gains to leverage). Based on our nine

D V

choices from 0.1 to 0.9, we can attempt to construct nine corresponding

βD’s and βL’s from which we can compute RD’s and RL’s needed in our GL formulation. We begin by estimating debt betas (βD’s). The estimation process is based on the observation that we have two βD’s endpoints and a βD interior point. This is seen below. For the 1st endpoint when

D V

= 0, we have βD = 0.

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For the interior point when For the 2nd endpoint when

D V = 0.5, we have (from Table D V  1.0, we have βD  βU =

2) βD = 0.2977. 0.7377.

Concerning the latter, we see that as a firm approaches an all-debt situation with all shares retired, it must legally revert to an all-equity firm and the unlevered beta of 0.7377. Using linear interpolation, we can estimate the βD’s for each book

D V

choice from 0.1 to 0.9. We

can then use the CAPM to get each corresponding RD. We show the βD and RD values below.

Debt Betas & Costs of Debt for Book βD RD

D V

Choices from 0.1 to 0.9

0.1 0.058

0.2 0.117

0.3 0.175

0.4 0.233

0.5 0.292

0.6 0.381

0.7 0.470

0.8 0.559

0.9 0.649

6.01%

6.36%

6.71%

7.06%

7.41%

7.94%

8.48%

9.02%

9.56%

D

D

For βL’s, we have only one endpoint ( V = 0, βL = βU = 0.7377) and an interior point ( V = 0.5, βL = 1.05), ruling out linear interpolation for all choices. Given βD’s and βU, we use Hamada (1972) to estimate βL’s. However, estimates using this equation break down as we approach high debt levels because the βL’s values generate the same small linear increase of 0.06245 for each successive book D V

choice from 0.6 to 0.9. Because of its linear relationship that treats the latter incremental increases

in debt as equally risky, the Hamada equation clearly cannot accommodate any expected rapidly increasing levels of financial risk as we reach extreme debt levels. Thus, for the last two choices, our application uses βL’s of 1.42 and 2.00 instead of the linear values given by Hamada (1.23735 and 1.29980). βL’s of 1.42 and 2.00 represent an increase of about 20% and 40% over respective previous βL’s, and are deemed a more acceptable attempt to capture the increasing levels of financial risk. The decision to start increasing βL’s for the 8th and 9th debt level choice is consistent with the target leverage ratio that is set for AGL Co. so as to avoid undue financial distress. It remains for future research to explore if there can be found a formula for βL supporting the observation that firms do not strive for extreme high levels of debt. Below we display the βL’s and RL’s for the book

D V

choices from 0.1 to 0.9 with the CAPM used

to compute RL’s.

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Equity Betas & Costs of Equity for Book βL RL

D V

Choices from 0.1 to 0.9

0.1 0.800

0.2 0.863

0.3 0.925

0.4 0.988

0.5 1.050

0.6 1.112

0.7 1.174

0.8 1.420

0.9 2.000

10.47%

10.84%

11.21%

11.59%

11.96%

12.34%

12.71%

14.18%

17.66%

C. Data Related to Unlevered Situation and Market Target DE for Application Given RL’s and RD’s for the nine debt-equity choices, we can get EU and EL by estimating values for C, γ TL arg et , PU, I, δg, and γU. We estimate the perpetual before-tax cash flow for unlevered equity owners (i.e., we estimate C) from http://www.stateone.com.au/documents/research/agl.pdf, which gives earnings before interest and taxes plus depreciation / amortization (EBITDA). For the years from 2003 to 2006, the average EBITDA suggests that C = $905.2 million. Next, we estimate the growth rate for an unlevered situation (γU). We begin by trying to determine current dollar change in after-tax cash flows (δg) as given by (9) where δg = (1 TPE)(1 TC)(C I)(γL) with γL defined as the levered growth rate when the firm achieves its targeted amount of interest paid (e.g., γL = γ TL arg et ). Noting that KPMG indicates that AGL Co.’s target

D E

ratio is 1.5 (which we take as the market ratio since KPMG uses the word ―market‖ when

describing the weights from this choice), we proceed to estimate the future growth rate for after-tax cash flows given this target. We settle on γ TL arg et = 5.4% .This estimate is consistent with data suggested by State One and AGL Co. reports. For example, State One (December 2004) estimates that the net profit after tax will change from $320.8 million for June 2003 to $370.4 million for June $376.4 2006. The implied growth rate in after-tax cash flows = γL = $320.8

1 3

1 ≈ 5.5%. A similar value is

found (γL ≈ 5.3%) if we take the average of the growth in dividends from June 2003 to June 2006 and the standard formula where the growth rate equals retention rate times required rate of return. To proceed with estimating δg, we next need to compute the interest paid (I) when γL = 5.4%. As will be seen later in Table 5, it is for the 7th debt level choice when 70% of unlevered shares are retired (book choice of

D V

= 0.7) that we attain the market target

D E

D

≈ 1.5 ( E ≈ 2.33 from a share

standpoint). As we will show later, I = $493,093,903 for this 7th choice. Using equation (9), Table 3 62 Northwest Missouri State University

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shows that δg = $14,834,558. This estimate is reasonably close to State One’s average increase in NPAT for 2003 2006 of about $15.7 million if adjusted for personal taxes. Given this value, we can now use (10) to solve for γU where we obtain γU ≈ 2.46%. Given γU, we can proceed to compute the unlevered equity value (EU). Table 3 shows that EU = $7,906,124,561. On a before personal tax basis, EU = $8,302,136,471. Dividing this by the number of unlevered shares (NU), we compute the stock price for the unlevered situation (PU) and get about $9.10 as shown in Table 3.

Table 3. Data Related to Unlevered Situation and Target Market D E NL = Number of Shares when Levered with Current Book NU = Number of Shares when Unlevered =

NL Current Book

E V

D E

=

is 1.0 = 456,000,000 456,000,000 = 912,000,000 0.5

C = Estimated by Average EBITDA (2003 2006) = $905,200,000 Target Market

D E

= Suggested Market Target Debt-Equity Choice = 1.5

γL = γ TL arg et = Estimated Growth Rate in After-Tax Cash Flows for Target Market

D E

= 5.4%

I = Interest Paid for Targeted Levered Situation = $493,093,903 δg = (1 TPE)(1 TC)(C I) γ TL arg et = (1 0.0477)(1 0.3)($905,200,000 $493,093,903)0.054 = $14,834,558 γU =

δg (1 TPE )(1 TC )C

=

$14,834,558.46 = 2.458432% ≈ 2.46% (1 0.0477 )(1 0.3)$905,200,000

EU (After Personal Taxes) =

(1 TPE )(1 TC )C (1 0.0477 )(1 0.3)$ 905,200,000 = = $7,906,124,561 0.100907 0.02458432 R U γU

EU (Before Personal Taxes) =

$7,906,124,561 $7,906,124,561 = = $8,302,136,471 (1 TPE ) (1 0.0477 )

PU (Per Share Unlevered Market Price) =

EU $8,302,136,471 = = $9.10322 NU 912,000,000

D. Company Data Related to Market Debt-Equity Target for Application Given δg and PU, we can now compute the interest paid (I) for each debt level choice given that I = RD(D) where as described earlier D = PU(SR). Because we have unlevered the firm where NU = 63 Northwest Missouri State University

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912,000,000 and PU = $9.10322, we can view each debt level choice for our unlevered firm as D = D

PU(SR) where SR is the number of shares being retired for that choice. Since SR = ( V )(NU), we have D

D

D = PU( V )(NU). Inserting this expression for D into I = RD(D), we get I = RD(PU)( V )(NU). Although the details are omitted, we can note that from this relationship, we solve for the earlier value for I = $493,093,903 for the 7th debt level choice. This is because we can create a quadratic equation where I is a function of TC, TPE, C, RD, RU, γ TL arg et , NU, and SR, all of which are known.

Table 4. Company Data Related to Market Debt-Equity Target (Values for Market Debt-Equity Choice = 1.5 which is Book Debt-Firm Value Choice = 0.7) R = Unlevered Shares Exchanged =

D V

NU = 0.7(912,000,000) = 638,400,000 shares

Book VD = Book Value Leverage Choice Given by Shares Retired =

R 638,400,000 = = 0.7 912,000,000 NU

D (Before Personal Tax) = Amount of EU Retired = PU(R) = $9.10322(638,400,000) = $5,811,495,529 I (Interest Paid) = RDD = 0.084848($5,811,495,529) = $493,093,903 EL = D=

(1 TPE )(1 TC )( C I) (1 0.0477 )(1 0.3)($ 905,200,000 $493,093,903) = = $3,756,195,004 0.12713626 0.054 R L γL

(1 TPD )I RD

=

Target Market

(1 0.2034 )$493,093, 903 = $4,629,437,339 0.08484802 D E

(On Before Personal Tax Basis) =

GL using (12) = 1

RU αR D D+ R L γL RL

1st Component = 1 2nd Component =

RU RL

γU γL

$4,629,437,339 / (1 0.2034 ) = 1.473 ≈ 1.5 $3,756,195,004 / (1 0.0477 )

1 EU = $135,068,376 + $344,439,406 = $479,507,782

αR D (0.836819 )( 0.08484802 ) D= 1 $4,629,437,339 = $135,068,376 0.12713626 0.054 R L γL γU γL

1 EU =

0.10090685 0.024584323 0.12713626 0.054

1 $7,906,124,561 = $344,439,406

VL = EL + D = $3,756,195,004 + $4,629,437,339 = $8,385,632,343 GL using (4) = VL – VU = $8,385,632,343 – $7,906,124,561 = $479,507,782 64 Northwest Missouri State University

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With our firm unlevered, we can now illustrate the computation I for any debt level choice from this unlevered condition, which is the condition assumed to apply when using (12). Consider the 1st choice where book

D V

D

= 0.1 and RD = 6.0142%. We have: I = RD (PU)( V )(NU) =

0.06014214($9.10322)(0.1)(912,000,000) = $49,930,825. Similarly, we can compute I for all choices. For the 7th choice, we can verify that I = $493,093,903 for book

D V

= 0.7. We have: I = RD

D

(PU)( V )(NU) = 0.084848($9.10322)(0.7)(912,000,000) = $493,093,903. As seen in Table 4, this interest payment corresponds very closely to the target market debt-equity choice of 1.5. After computing I for each debt level choice, we can use (11) and compute γL for each choice. For example, for the 1st debt level choice, we get γL =

δg (1 TPE )(1 TC )( C I)

=

$14,834,55 8.46 = 2.601956%. (1 0.0477 )(1 0.3)($ 905,200,000 $49,930,825)

We obtain the following γL values for the nine debt level choices: 2.602%, 2.783%, 3.016%, 3.318%, 3.725%, 4.370%, 5.400%, 7.270%, and 11.637%. The growth rates begin increasing much more rapidly as I  C causing the 1st and 2nd components of (12) to eventually flip signs. Given γL for each debt level choice, we can use (12) or (4) to get GL for each corresponding debt level choice. We do this in Table 4 for the 7th debt level choice and show that GL = $479,507,782 on an after personal tax basis. The table also shows that this choice corresponds with a target market

D E

≈ 1.5 when computed on a before personal tax basis.

V. Results for All Nine Debt Level Choices for the Application This section summarizes the results of our application in table form. We give gain to leverage (GL) results for the unlevered application for AGL Co. for all nine debt level choices. The application assumes the previously mentioned data including the betas needed to compute the costs of capital. Conditions of our application are formally stated below so as to include values for key variables. (a) debt is risky with RD > RF = 5.6642%, and RD is positively related to debt. (b) tax rates are relevant with TPE = 4.77%, TPD = 20.34%, and TC = 30%; (c) uncertain perpetual before-tax cash flows to unlevered equity: C = $905,200,000; D

(d) constant growth rate when target market E approximated: γ TL arg et = 5.4% with current dollar change (δg) = $14,834,558; and, (e) an unlevered firm with risky equity faces a finite set of perpetual debt-for-equity choices with RL > RU = 10.0907%. 65 Northwest Missouri State University

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Table 5. Application of Gain to Leverage Formulation for a Real World Firm Assuming Risky Debt, Personal Taxes, and Constant Growth Rate Panel A. On After Personal Tax Basis with Currency in Billions of Australian Dollars Book D V

RD

RL

1st 2nd Component Component

Market GL

D

EL

VL

D E

0.1

0.0601 0.1047

0.2381

0.2324

0.0056 0.661 7.250 7.912

0.091

0.2

0.0636 0.1084

0.4484

0.4165

0.0318 1.323 6.615 7.938

0.200

0.3

0.0671 0.1121

0.6245

0.5467

0.0777 1.984 6.000 7.984

0.331

0.4

0.0706 0.1159

0.7547

0.6106

0.1440 2.645 5.405 8.050

0.489

0.5

0.0741 0.1196

0.8167

0.5825

0.2341 3.307 4.834 8.140

0.684

0.6

0.0795 0.1234

0.6555

0.3338

0.3217 3.968 4.260 8.228

0.932

0.7

0.0848 0.1271

0.1351

0.3444

0.4795 4.629 3.756 8.386

1.232

0.8

0.0902 0.1418

0.4850

0.8208

0.3358 5.291 2.951 8.246

1.793

0.9

0.0956 0.1766

1.9447

2.1058

0.1611 5.952 2.115 8.067

2.814

Panel B. On Before Personal Tax Basis with Currency in Billions of Australian Dollars Book D V

RD

RL

1st 2nd Component Component

Market GL

D

EL

VL

D E

0.1

0.0601 0.1047

0.3857

0.2441

0.1417 0.830 7.614 8.444

0.109

0.2

0.0636 0.1084

0.7422

0.4374

0.3049 1.660 6.947 8.607

0.239

0.3

0.0671 0.1121

1.0630

0.5741

0.4888 2.491 6.300 8.791

0.395

0.4

0.0706 0.1159

1.3354

0.6412

0.6942 3.321 5.676 8.996

0.585

0.5

0.0741 0.1196

1.5363

0.6117

0.9246 4.151 5.076 9.227

0.818

0.6

0.0795 0.1234

1.5028

0.3505

1.1523 4.981 4.473 9.454

1.114

0.7

0.0848 0.1271

1.0920

0.36170

1.4537 5.811 3.944 9.756

1.473

0.8

0.0902 0.1418

0.5767

0.8619

1.4386 6.642 3.099 9.741

2.143

0.9

0.0956 0.1766

0.8205

2.2113

1.3908 7.472 2.221 9.693

3.364

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Table 5 reports results for ten key variables when financial managers face nine debt choices given in the nine rows. Panel A reports results (where applicable) on an after personal tax basis while Panel B focuses on before personal tax results. As discussed previously, the before personal tax results are important because they can minimize inaccuracies in the after personal tax results that might result if personal tax rates are closer in value than what we use. Each panel has two bold-faced rows. The 1st bold-faced row is for the current situation where D

book V = 0.5, while the 2nd bold-faced row is for book

D V

= 0.7, which is where GL is maximized for

both panels. As seen in the last column of Panel B, it is also the row which is nearest the market target

D E

of 1.5. For this row, we get GL = $1.4537 billion on a before personal tax basis (which is

what the market sees). For this row, dividing EL by the number of outstanding shares (NL), we get a share price ≈ $14.42. For example, with

D V

= 0.7 (or VE = 0.3), we have NL = VE (NU) =

0.3(912,000,000) = 273,600,000 shares giving the market share price as: EL $3,944 ,340 ,023 = = $14.4164 per share. 273,600 ,000 shares NL

PBefore Personal Tax =

This is less than the average market price at the time of this writing, which has averaged $13.83 for January 2005. Thus, $14.42 can be considered a prediction of the future price (absent effects beyond those stemming from the increased debt) if the market target is achieved. The prediction for the stock price at the time we begin estimating values for our variables (February 2004) can be computed for the 1st bold-faced row where NL = 456,000,000 shares. We have: PBefore Personal Tax =

EL $5,075,687 ,792 = = $11.1309 per share. 456,000,000 shares NL

This price is consistent with both the average price of $11.06 for AGL Co. for February 2004 and also for the average price of $11.29 for the year of the 2003 annual report (7/1/03 to 6/30/04). GL on a before personal tax basis in Panel B is greater than the after personal tax basis in Panel A given that personal taxes are subtracted from GL in Panel A. The difference is sizeable as seen when comparing the maximum GL value of $0.4795 billion in Panel A with the corresponding maximum value of $1.4537 billion in Panel B. In looking at Panel B, we can also point out, that due 67 Northwest Missouri State University

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to the increase in equity that accrues from GL, the book reduced to a market

D E

D E

of 2.333 (from a share standpoint) is

of 1.473. This is near the target of 1.5.

While the maximum GL is achieved in both panels with the 7th debt level choice, it does not necessarily follow that both panels will agree. It may be even more likely that the after personal tax GL can be achieved for a different debt level choice than the before personal GL. If decisions are actually made on what is best for investors, the firm would arguably choose the debt level where G L is maximized on an after personal tax basis. However, absent perfect knowledge about personal tax rates and given our restriction to nine debt level choices (where this restriction tends to underestimates the maximum GL and the optimal before personal tax basis will occur within a

D E

D E

), it appears that the firm’s maximum GL on a

range of about 1.4 to 2.0. This range is consistent

with values reported by http://www.bizstats.com/currentratios.htm. For example, BizStats give a debt-to-equity ratio of 1.79 for U.S.A. gas production and distribution utilities. Although both GL components experience change in signs as more debt is added, this is not necessarily always the case. Absent a large value for δg that leads to large values for γL for high debt levels, we would expect the 1st component to always be positive and the 2nd component to always be negative.

VI. Limitations of Application In this section, we call attention to four limitations facing our application. In general, such limitations are found in all models that rely on accurate estimates of values for its variables. First, personal tax rates were not directly known. This limitation was ameliorated through use of an effective tax rate and analysis of before personal tax values. Second, most firms are levered. Thus, to apply our GL formulation, we have to unlever our firm in an attempt to estimate the number of shares outstanding (NU) if it had no debt. From here we determine book debt-to-equity choices. Given these choices and the unlevered price (PU), we can determine how much debt will be issued for each choice. The application depends on the accuracy of estimating NU and PU and is limited if these estimates lack sufficient accuracy. Third, we encountered a roadblock when computing betas. For example, we had to interpolate from endpoints and a midpoint to get reasonable βD’s for each debt level choice. From there we 68 Northwest Missouri State University

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proceeded to get βU and then obtain βL’s for the nine debt level choices by using standard formulas. However, unless adjusted upward, the βL computations for higher debt levels would suggest that firms aim for extremely high leverage targets, which we do not find in the real world. This limitation (in getting sufficient estimates for at least some betas) caused us to make intuitive ad hoc assignments for levered equity betas at higher levels of debt. Future research needs to explore other ways of estimating betas and costs of capital such as suggested by researchers who offer alternatives methods (Fama and French, 1997; Lally, 2004). Finally, the application had to estimate a current dollar level of growth (δg) based on a chosen growth rate at the target debt-equity choice. Using Excel, we were able to solve for δg and the interest paid (I) at the market target debt-equity choice based on values for other variables.

VII. Summary Statements This paper derives GL formulations based on definitions for unlevered and levered firm values. Such formulations include discount rates for unlevered equity, levered equity, and debt. The inclusion of these rates makes it possible for GL values to eventually decrease with increasing debt levels. Three GL formulations for an unlevered situation are offered to aid managers (when making the debt-equity choice) and educators (when explaining the ramifications of the debt-equity choice). The application using market data and company data for AGL Co. showed how managers can use the GL formulation with personal taxes and constant growth to understand how the debt-equity choice can influence firm value. While this paper’s model (like any model) relies on accurate estimates of values for variables, the model’s optimal GL was able to conform to the recommended market target

D E

of 1.5 by assuming escalating values for levered equity’s beta at higher levels of

debt. This study is important for several reasons. First, prior research offers formulations that are difficult for practitioners in that they do not fully address the role of discount rates, and tend to be unrealistic by including variables that are virtually immeasurable in themselves (e.g., direct and indirect bankruptcy costs and agency costs). As such, financial managers are hard pressed to find utility in their application. To the extent changes in discount rates are easier to estimate, this paper's GL formulations offer more practical potential. Second, the practical application in this paper suggests a wealth maximizing debt-equity choice 69 Northwest Missouri State University

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where the actual choice can depends on taxes and growth rates in addition to changes in discount rates. The application produces results that re-enforce and strengthen general conclusions of prior empirical and theoretical research in regard to the belief that increasing levels of debt can cause G L to begin falling. The increase in G L, followed by a decrease, can be explained through the interaction of many factors (such as taxes, bankruptcy costs, and agency effects) that together determine a firm’s optimal debt-equity choice. In conclusion, the GL formulations found in this paper reaffirm, synthesize, and extend prior GL formulations, while opening up a fresh vista from which to view the debt-equity choice faced by managers. This vista offers a practical vantage point in that capital structure decision-making can be based on formulations that include variables heretofore not fully utilized.

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Appendix A: GL for Unlevered Firm with No Personal Taxes Proof of equation (8): For the situation of an unlevered firm when only corporate taxes are considered, substituting (7) into (4) and noting VU = EU gives GL =

(1 TC )( C I) I + RL RD

Multiplying out the 1st component, noting GL = D Multiplying the 2nd component by

EU.

I = D, and rearranging: RD

(1 TC )I RL

RD gives RD

EU +

(1 TC )

(1 TC )C . RL

RD I R , which is (1 TC ) D D, and RL RD RL

factoring out D: GL = 1 (1 TC )

Multiplying the last component by

RD RL

RU RU gives RL RU

D

EU +

(1 TC )C . RL

RU (1 TC )C , which is EU, and factoring RU RL

out EU: GL = 1 (1 TC ) Setting α = (1 TC) and noting

1

RD RL

D

RU RU EU = + RL RL

GL = 1

1

RU EU. RL

1 EU gives

RU αR D D+ RL RL

1 EU.

(8)

Q.E.D.

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Appendix B: GL for Unlevered Firm with Personal Taxes and Constant Growth Proof of equation (12): Assume constant growth such that γL > γU > 0 and personal taxes such that VU = EU =

(1 TPE )(1 TC )C (1 TPE )(1 TC )( C I) and EL = . Substituting VL = EL + D = R U γU R L γL

(1 TPE )(1 TC )( C I) + D into (4) and noting VU = EU gives: R L γL (1 TPE )(1 TC )( C I) +D R L γL

GL =

EU.

Multiplying out the 1st component and rearranging: (1 TPE )(1 TC )I R L γL

GL = D

Multiplying the 2nd component by

which is

(1 TPE )(1 TC ) (1 TPD )

GL = 1

RU RL

RD RL

γL

(1 TPD )I , RD

D, and factoring out D:

γL

(1 TPE )(1 TC ) (1 TPD )

Multiplying the last component by

(1 TPE )(1 TC )C . R L γL

(1 TPE )(1 TC ) (1 TPD )

(1 TPD )R D gives (1 TPD )R D

RD RL

EU +

RU RU

RD RL

D

γL

(1 TPE )(1 TC )C . R L γL

EU +

RU γU = 1 gives RL γU

γU γL

(1 TPE )(1 TC )C , which is R U γU

γU EU, and factoring out EU: γL

GL = 1 Setting α =

(1 TPE )(1 TC ) (1 TPD )

(1 TPE )(1 TC ) and noting (1 TPD )

GL = 1

RD RL

1

RU RL

γL

D

1

RU RL

γU RU EU = + γL RL

RU αR D D+ R L γL RL

γU γL

γU EU. γL γU γL

1 EU gives

1 EU.

Q.E.D. 72 Northwest Missouri State University

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(12)

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